Running a crude model using Bayesian inference, I get some results > 1 (ie, more than 100% "certain") for some combinations of "evidence". For instance, for one bit of evidence the conditional probability of the null hypothesis is 0.85 while the marginal probability is 0.77. If the prior probability is 0.9, the computed posterior probability is 1.008. Which maybe could be ascribed to rounding error, except that the next bit of evidence raises the posterior probability to 1.34.
It stands to reason that, for a problem with two hypotheses, one would have a conditional probability less than the marginal probability and the other would be greater. So the resulting P(E|H) / P(E) multiplier would be > 1. So it's hard to see how such > 1 results can be avoided in the general case.
Is this just the way Bayesian inference works, or do I likely have an error somewhere in my calcs?
One "evidence": Total 6134 samples Total actual positives 2845 Total actual negatives 3289 Test was true 1623 times Test was true 465 times when the "gold standard" was positive Test was true 1158 times when the "gold standard" was negative conditional probability of positive hypothesis = 465/2845 = 0.1634 conditional probability of negative hypothesis = 1158/3289 = 0.3521 marginal probability of test being true = 1623/6134 = 0.2646 Bayes multiplier for the negative hypothesis = 0.3521 / 0.2646 = 1.3307
As can be seen, the multiplier is significantly > 1, and with several such tests back-to-back it seems hard to avoid probabilities > 1. (Of course, I suppose one can argue that the tests aren't truly independent, and that puts the fly in the ointment.)
So does anyone have any suggestions as to how to "fudge" non-independent measurements to improve an estimate? For the two most egregious cases I can come up with a fair estimate of how connected the measurements are, but I don't have a feel for how to factor that knowledge in.